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Bài tập Toán DIFFERENTIATION OPTIMIZATION 08
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Created by T. Madas
Question 20 (****)
The figure below shows the design of an animal feeder which in the shape of a hollow,
open topped half cylinder, made of thin sheet metal. The radius of the semicircular ends
is r cm and the length of the feeder is L cm.
The metal used in the construction of the feeder is 600π cm 2 .
L
r
a) Show that the capacity, V cm3 , of the feeder is given by
V = 300π r − 1 π r 3 .
2
The design of the feeder is such so its capacity is maximum.
b) Determine the exact value of r for which V is stationary.
c) Show that the value of r found in part (b) gives the maximum value for V .
d) Find, in exact form, the capacity and the length of the feeder.
r = 10 2 ≈ 14.14 , L = 20 2 ≈ 28.28 , Vmax = 2000π 2 ≈ 8886
Created by T. Madas
Created by T. Madas
Question 21 (****)
y
2x
The figure above shows the design of a window which is the shape of a semicircle
attached to rectangle. The diameter of the semicircle is 2x m and is attached to one side
of the rectangle also measuring 2x m . The other side of the rectangle is y m .
The perimeter of the window is 6 m .
a) Show that the total area of the window, A m 2 , is given by
A = 6 x − 1 ( 4 + π ) x2 .
2
b) Given that the measurements of the window are such so that A is maximum,
show by a method involving differentiation that this maximum value of A is
18
.
4 +π
proof
Created by T. Madas
Created by T. Madas
Question 22 (****)
The figure below shows the design of a hazard warning logo which consists of three
identical sectors of radius r cm, joined together at the centre.
Each sector subtends an angle θ radians at the centre and the sectors are equally spaced
so that the logo has rotational symmetry of order 3 .
θ
r
The area of the logo is 75 cm 2 .
a) Show that the perimeter P cm of the logo is given by
P = 6r +
150
.
r
b) Determine by differentiation the value of r for which P is stationary.
c) Show that the value of r found in part (b) gives the minimum value for P .
d) Find the minimum perimeter of the feeder.
r = 5 , Pmin = 60
Created by T. Madas
